The geometric sequence $a_i$ is defined by the formula: $a_1 = 10$ $a_i = a_{i - 1} \cdot \dfrac{9}{10}$ Find the sum of the first $75$ terms in the sequence. Choose 1 answer: Choose 1 answer: (Choice A) A $99.96$ (Choice B) B $100.00$ (Choice C) C $112.61$ (Choice D) D $1 \cdot 10^{74}$
Answer: Getting started Let's write out the first few terms of the series: $10 + 9 + 8.1...$ We're dealing with a geometric series because each term is multiplied by $\dfrac{9}{10}$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {10})$ and the number of terms $(n = {75})$ are given in the question. The common ratio $r$ is ${\dfrac{9}{10}}$ because each term is multiplied by ${\dfrac{9}{10}}$ to get the next term. Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{75}}&=\dfrac{{10}\left(1-\left({\dfrac{9}{10}}\right)^{{75}}\right)}{1-{\dfrac{9}{10}}} \\\\ S_{{75}}&=100\left(1-\left({\dfrac{9}{10}}\right)^{{75}}\right)\\\\ S_{{{75}}} &\approx 99.96\end{aligned}$ The answer $99.96$